Immutable quantized transport in Floquet chains

We show how quantized transport can be realized in Floquet chains through encapsulation of a chiral or helical shift. The resulting transport is immutable rather than topological in the sense that it neither requires a band gap nor is affected by arbitrarily strong perturbations. Transport is still characterized by topological quantities but encapsulation of the shift prevents topological phase transitions. To explore immutable transport we introduce the concept of a shiftbox, explain the relevant topological quantities both for momentum-space dispersions and real-space transport, and study model systems of Floquet chains with strictly quantized chiral and helical transport. Natural platforms for the experimental investigation of these scenarios are photonic Floquet chains constructed in waveguide arrays, as well as topolectrical or mechanical systems.

Figure 1

(a) Photonic waveguide implementation of one period of the Floquet chain from Ref. [35]. The vertical axis corresponds to time. (b) Quasienergy dispersion $\varepsilon \left(k\right)$ in the general case ($J\ne \pi /2$, solid curves) and at perfect coupling ($J=\pi /2$, dashed lines). (c) Photonic waveguide implementation of a helical shiftbox for the operator $U_{\leftrightarrow }$. (d) Quasienergy dispersion $\varepsilon \left(k\right)$ associated with the helical shiftbox.

Figure 2

(a) Photonic waveguide implementation of a chiral shiftbox for the operator $U_{\to }$. (d) Quasienergy dispersion $\varepsilon \left(k\right)$ associated with the chiral shiftbox.

Figure 3

(a) Chiral shiftbox sandwiched between standard Floquet dynamics. (b) Helical shiftbox sandwiched between standard Floquet dynamics with fermionic TRS. (c) Quasienergy dispersion $\varepsilon \left(k\right)$ for the sandwiched chiral shiftbox with $J=1$ (see text for definitions). (d) Quasienergy dispersion $\varepsilon \left(k\right)$ for the sandwiched helical shiftbox with $J=J^{\prime }=1$.

Figure 4

Bidirectional transport with the helical shiftbox. (a) Transport measured per pseudospin direction (${\overline{C}}_{\uparrow },{\overline{C}}_{\downarrow }$) is quantized only for uncoupled pseudospin components, that is for $J^{\prime }=0$ (see text). (b) Transport measured for Wannier functions (${\overline{C}}_r,{\overline{C}}_l$) is strictly quantized for all parameters.